the construction of free-free flexibility matrices

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CU-CAS-01-06 CENTER FOR AEROSPACE STRUCTURES

The Construction of

Free-Free Flexibility Matrices for

Multilevel Structural Analysis

by

C. A. Felippa and K. C. Park

June 2001 COLLEGE OF ENGINEERING

UNIVERSITY OF COLORADO

CAMPUS BOX 429

BOULDER, COLORADO 80309


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The Construction of Free-Free Flexibility Matrices

for Multilevel Structural Analysis


Department of Aerospace Engineering Sciences

and Center for Aerospace Structures

University of ColoradoBoulder, Colorado 80309-0429, USA

April 1998

Revised June 2001

Report No. CU-CAS-01-06

Research supported by the National Science Foundation under Grant ECS-9217394, and

by Livermore and Sandia National Laboratories under Contracts B347880 and AS-5666.

Submitted for publication to Computer Methods in Applied Mechanics and Engineering.


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THE CONSTRUCTION OF FREE-FREE FLEXIBILITY MATRICES

FOR MULTILEVEL STRUCTURAL ANALYSIS


Department of Aerospace Engineering Sciences and

Center for Aerospace Structures

University of Colorado

Boulder, Colorado 80309-0429, USA

SUMMARY

This article considers a generalization of the classical structural flexibility matrix. It expands on previous

papers by taking a deeper look at computational considerations at the substructure level. Direct or

indirect computation of flexibilities as influence coefficients has traditionally required pre-removal of

rigid body modes by imposing appropriate support conditions, mimicking experimental arrangements.

Withthe method presentedhere theflexibilityof an individual element or substructure is directly obtained

as a particular generalized inverse of the free-free stiffness matrix. This generalized inverse preserves

the stiffness spectrum. The definition is element independent and only involves access to the stiffness

generated by a standard finite element program and the separate construction of an orthonormal rigid-body mode basis. The free-free flexibility has proven useful in special application areas of finite element

structural analysis, notably massively parallel processing, model reduction and damage localization. It

can be computed by solving sets of linear equations and does not require processing an eigenproblem or

performing a singular value decomposition. If substructures contain thousands of degrees of freedom,

exploitation of the stiffness sparseness is important. For that case the paper presents a computation

procedure based on an exact penalty method, and a projected rank-regularized inverse stiffness with

diagonal entries inserted by the sparse factorization process. These entries can be physically interpreted

as penalty springs. This procedure takes advantage of the stiffness sparseness while forming the full

free-free flexibility, or a boundary subset, and is backed by an in-depth null space analysis for robustness.

Keywords: free-free flexibility, free-free stiffness, finite elements, generalized inverse, modified inverse,

projectors, structures, substructures, penalty springs, parallel processing.

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1 MOTIVATION

The direct or indirect computation of structural flexibilities as influence coefficients has traditionally

required precluding rigid body modes by imposing appropriate support conditions. This analytical

approach mimics time-honored experimental practices for static tests, in which forces are applied to a

supported structure, and deflections measured. See Figure 1 for a classical example.

Rigid blocks P

; ; ; ; ; ; ; ; ;

Figure 1. Old-fashioned experimental determination of wing flexibilities. The

aircraft is set on rigid blocks (not its landing gear) and a load P applied

at the wing tip torque-center. The ratio of tip deflection to P gives the

overall bending flexibility coefficient. The other key coefficientfor flutter speed calculations is the torsional flexibility, obtained

by measuring the pitch rotations produced by an offset tip load.

There are several application areas, however, for which it is convenient to have an expression for

the flexibility matrix of a free-free structure, substructure or element. Important applications include

domaindecomposition for FETI-type iterative parallel solvers [15], as well as systemidentificationand

damage localization [69]. The qualifier free-free is used here to denote that all rigid body motions

are unrestrained. This entity will be called a free-free flexibility matrix, and denoted by F.

In the symmetric case, which is the most important one in FEM modeling, the free-free flexibility

represents the true dual of the well known free-free stiffness matrix K. More specifically, F and K are

the pseudo-inverses (also called the Moore-Penrose generalized inverses) of each other. The generalexpression for the pseudo-inverse of a singular symmetric matrix such as K involves its singular value

decomposition (SVD) or, equivalently, knowledge of the eigensystem ofK; see e.g. Stewart and Sun

[10]. That kind of analysis is not only expensive, but notoriously sensitive to rank decisions when

carried out in floating-point arithmetic. That is, when can a small singular value be replaced by zero?

Such decisions depend on the problem and physical units, as well as the working computer precision.

Explicit expressions are presented here that relate K and F but involve only matrix inversions or,

equivalently, the solution of linear systems. These expressions assume the availability of a basis matrix

R for the rigid body modes (RBMs), which span the null space ofK. Often R may be constructed by

geometric arguments separately from K and F. Because no eigenanalysis is involed, the formulas are

suitable for symbolic manipulation in the case of simple individual elements.

The free-free flexibility hasbeen introducedin previous papers [11,12], but these have primarily focused

on theoretical and system level use considerations. Four practical aspects are studied here in more

detail at the individual substructure level:

1. Relations between the free-free boundary-reduced stiffness and flexibility matrices (Section 4).

2. Cleaning up RBM-polluted matrices by projector filtering (Section 5).

3. Use of congruential forms in symbolic processing of individual elements (Section 6).

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4. Ef ficient computation ofF for substructures of moderate or large size (Section 7). The procedure

relies on a general expression that involves a regularization matrix and the RBM projector.

The paper is organized as follows. Section 2 gives terminology and mathematical preliminaries. Sec-

tion 3 introduces the free-free flexibility and its properties. Section 4 discusses the reduction of these

expressions to boundary freedoms. Section 5 discusses projector filtering to clean up RBM-polluted

matrices. Section 6 derives the free-free flexibility of selected individual elements. Section 7 addressesthe issue of efficient computation ofF, given K and R, for arbitrary substructures. Section 8 summarizes

our key findings. The Appendix collects background material on generalized inverses and the inversion

of modified matrices.

2 SUBSTRUCTURES TERMINOLOGY

A substructure is defined here as an assembly offinite elements that does not possess zero-energy modes

aside from rigid body modes (RBM). See Figure 2. Mathematically, the null space of the substructure

stiffness contains only RBMs. (This restriction may be selectively relaxed, as further discussed in

Section 7.9). Substructures include individual elements and a complete structure as special cases. The

total number of degrees of freedom of the substructure is called NF.

A

(b)(a)

Figure 2. Whereas (a) depicts a legal 2D substructure, the assembly (b) contains a

non-RBM zero energy mode (relative rotation about point A). If such

mechanisms are disallowed, (b) is not a legal substructure.

Should it be necessary, substructures are identified by a superscript enclosed in brackets, for example

K[3] is the stiffness of substructure 3. Because this paper deals only with individual substructures, that

superscript will be omitted to reduce clutter.

Figure 3(a) depicts a two-dimensional substructure, and the force systems acting on its nodes from

external agents. The applied forces fa are given as data. The interaction forces are exerted by other

connected substructures. If the substructure is supported or partly supported, it is rendered free-free on

replacing the supports by reaction forces fs as shown in Figure 3(b). The total force vector acting on

the substructure is the superposition

f= fa + fs + (1)where each vector has length NF. Vectors and fs are completed with zero entries as appropriate for

conformity.

At each node considered as a free body, the internal force p is defined to be the resultant of the acting

forces, as depicted in Figure 3(c). Hence the statement of node by node equilibrium is:

f p = 0, or fa + fs + p = 0, (2)

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;

;

;

;

Applied forces (those at interior

nodes not shown to reduce clutter)

Interaction forces

Support reaction forces

Internal forces, only shown in (c)

f

[s]

ff

[s]

(b)

(c)

(a)

j

p

s

a

x

y

z

Figure 3. A generic substructure [s] and the force systems acting on it. The top figures

illustrate the conversion from a partly or fully supported configuration (a)

to a free-free configuration (b) by replacing supports by reaction forces.

Diagram (c) illustrates the self-equilibrium of a free-body node j .

For some applications, notably iterative solvers, it is natural to view support reactions as interaction

forces by regarding the ground as another substructure (often identified by a zero number). In such

case vector fs is merged with .

As for the kinematics, the free-free substructure has NR > 0, linearly independent rigid body modes

or RBMs. Rigid motions are characterized through the RBM-basis matrix R, dimensioned NF NR ,such that any rigid node displacement can be represented as uR = Ra where a is a vector ofNR modalamplitudes. The total displacement vector u can be written as a superposition of deformational and

rigid motions:

u = d + uR = d + Ra, (3)Matrix R may be constructed by taking as columns NR linearly independent rigid displacement fields

evaluated at the nodes. For conveniency those columns are assumed to be orthonormalizedso that

RTR = IR , the identitymatrixof order NR . Theorthogonalprojector associated with R is thesymmetricidempotent matrix

P = I R(RTR)1RT = I RRT, (4)where I (without subscript) is the NF NF identity matrix. The last simplification in (4) arises fromthe assumed orthonormality ofR. Note that P2 = P and PR = RTP = 0.Applicationof theprojector (4) to u extracts the deformational node displacements d = Pu. Subtractingthese from the total displacements yields the rigid motions uR = u d = (I P)u = RRTu, whichshows that a = RTu. Using the idempotent property P2 = P it is easy toverify that dTuR = 0. Figure4gives a geometric interpretation of the decomposition (3).

Invoking the Principle of Virtual Work for the substructure under virtual rigid motions uR = R a

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u

d = u Ra = (I R R ) u = P uT

u = R R u = R aTR

Rigidbodymotions

(a.k.a.nullspace)

Deformational motions

(a.k.a. range space)

Figure 4. Geometric interpretation of the projector P.

yields

RTf= RT(fa + fs + ) = 0, (5)as the statement of overall self-equilibrium for the substructure.

3 STIFFNESS AND FLEXIBILITY MATRICES

3.1 Definitions

The case of real-symmetric stiffness and flexibility matrices is the most important one in practice.

This article only considers that case. The real-unsymmetric case, which arises in nonconservative and

corotational FEM models, is briefly discussed in [11].

The well known free-free stiffness matrix K of a linearly elastic substructure relates node displacements

to node forces through the stiffness equations:

K u

=f

=fa

+fs

+, K

=KT. (6)

We assume throughout that K is nonnegative. IfK models all rigid motions exactly, RTK = KR mustvanish identically on account of (5). Such a stiffness matrix will be called clean as regards rigid body

motions or, briefly, RBM-clean.

The free-free flexibility F of the substructure is defined by the expressions

F = P (K+ RRT)1 = (K+ RRT)1 P = FT. (7)

The symmetry ofF follows from the spectral properties discussed in the next subsection. Using F we

can write the free-free flexibility matrix equation dual to (6) as

F f= d = Pu = u uR. (8)

Premultiplication by RT and use of (5) shows that FR = 0.If the substructure is fixed (that is, fully restrained against all rigid motions), matrix R is void. The

definition (7) then collapses to that of the ordinary structural flexibility F = K1 whereas (8) reducesto F f= u uR = u. Because of coalescence in the fully supported case, the same matrix symbol Fcan be used without risk of confusion.

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3.2 Spectral and Duality Properties

The basic properties can be expressed in spectral language as follows. The free-free stiffness K has

two kind of eigenvalues:

1. NR zero eigenvalues pertaining to rigid motions. The associated null eigenspace is spanned by the

columns ofR, because that basis matrix is assumed to be orthonormal.

2. Nd = NF NR nonzero eigenvalues i . The associated orthonormalized deformational eigen-vectors vi , which span the range space ofK, satisfy Kvi = i vi , RTvi = 0 and vTi vj = i j .

The eigenvectors ofK + RRT are identical to those ofK but the RBM eigenvalues are raised to unity,giving the spectral decompositions

K + RRT =Ndi=1

i vi vTi +RRT, (K+ RRT)1 =

Ndi=1

1

ivi v

Ti +RRT. (9)

By construction the projector P = IRRT

has Nd unit eigenvalues whose range eigenspace is spannedby the vi , and the same null eigenspace as K. Consequently, use of orthogonality properties yields the

spectral decompositions

P =Ndi=1

vi vTi , F = P(K+ RRT)1 =

Ndi=1

1

ivi v

Ti . (10)

The foregoing relations show that the three symmetric matrices K, F and P have the same eigenvectors,

and can be commuted at will. For example, FK P = KPF for any integer exponents ( , , ).Commutativity proves the symmetrization in (7). Other useful relations that emanate from the spectral

decompositions are

K = P(F +RRT)1 = (F +RRT)1P = KT, (11)

(K+ RRT)(F + RRT) = I + ( 1)RRT = P + RRT (, arbitrary scalars), (12)

KR = FR = PR = 0, (13)

KFK = K, FKF = F, (KF)T = FK = KF = P, (FK)T = KF = FK = P, (14)

RT(K+ RRT)1 R = I, RT(F +RRT)1 R = I. (cf. Section A.3.2) (15)This relation catalog shows that K and F are dual, because exchanging them leaves all formulas intact.

Comparing to the definitions in Section A.1 of the Appendix it is seen that F and K are both pseudo-inverses and sg-inverses of each other: F = K+ = K and K = F+ = F. The practical importanceof (7) is that ifK and R are given, F can be computed by solving linear equations without need of

the more expensive eigenvalue analysis ofK. This is important for substructures containing hundreds

or thousands of elements because linear equation solvers can take advantage of the natural sparsity of

K, as discussed in Section 7. The stiffness matrix can be efficiently generated by the Direct Stiffness

Method using existing finite element libraries, whereas the construction ofR from geometric arguments

is straightforward as explained in Section 3.4.

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3.3 Alternative Expressions

The flexibility expressions (8) are actually the first two of the following 12 formulas for F,

P(K +RRT)1, (K+ RRT)1 P, P(K+ RRT)1 P,P(PK

+RRT)1, (PK

+RRT)1 P, P(PK

+RRT)1 P,

P(KP +RRT)1, (KP +RRT)1 P, P(KP + RRT)1 P,P(PKP +RRT)1, (PKP +RRT)1 P, P(PKP +RRT)1 P.

(16)

The seventh form, P(KP + RRT)1, was found by Bott and Duffin [13] in a different context. Ex-pressions (16) are equivalent in exact arithmetic ifK is RBM clean in the sense that KR = 0. IfK is, however, polluted so that KR = 0 the expressions (16) will generally yield different resultsfor F. Furthermore, matrices given by the formulas in the first three rows may not be symmetric. If

K is polluted, the last three formulas are recommended, because the filtered stiffness K = PKP isguaranteed to be both symmetric and clean. This point is further examined in Section 5.

Similarly, ifF is known, K may be computed from one of the 12 formulas

P(F+ RRT)1, (F +RRT)1 P, P(F+ RRT)1 P,P(PF + RRT)1, (PF + RRT)1 P, P(PF + RRT)1 P,P(FP + RRT)1, (FP + RRT)1 P, P(FP + RRT)1 P,

P(PFP + RRT)1, (PFP + RRT)1 P, P(PFP + RRT)1 P,

(17)

which are equivalent ifFR = 0.Using the fact that Pk = P for any integer k > 0, any P in (16) and (17) may be raised to arbitraryinteger powers, but this generalization is vacuous.

3.4 Forming the RBM Matrix

If the substructure is free-free and has no spurious modes, the construction ofR by geometric arguments

is straightforward. This is illustratedhere for thetwo-dimensionalcase depicted inFigure3. To facilitate

satisfaction of orthogonality, it is convenient to place the x, y axes at the geometric mean of the Nnode

locations of the substructure. Three independent RBMs are the x translation ux = 1, uy = 0, the ytranslation ux = 0, uy = 1 and the z rotation ux = y, uy = x . Evaluation at the nodes gives

RT =

1 0 1 . . . 0

0 1 0 . . . 1

y1 x1 y2 . . . xN

. (18)

The columns of this R are mutually orthogonal by construction. All that remains is to normalize them tounit length through division by N1/2, N1/2 and [

i (x

2i +y2i )]1/2, respectively. The three-dimensional

case is equally straightforward if thesubstructure hasonly translational freedoms. If rotational freedoms

are present, explicit orthonormalization using Gram-Schmid or similar scheme is required.

In geometrically nonlinear Lagrangian analysis a common occurrence is the loss of rotational rigid body

modes dues to initial stress effects in the geometric stiffness, which results in a gain of rank ofK with

respect to the linear case. In plasticity analysis a loss of rank due to plastic flow mechanisms may occur.

In either case the null space basis ofK has to be appropriately adjusted.

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; ; ;

; ; ;

; ; ;

; ; ;

; ; ;

; ; ;

; ; ;

; ; ;

; ; ;

; ; ;

; ; ;

(a) (b) (c)

Figure 5. Reduction to boundary freedoms in the domain decomposition of a FEM mesh. In solving

the so-called coarse problem, which involves substructural interactions, only boundary

node freedoms participate. The rightmost figure depicts such interactions occurring

indirectly through intermediate connection frames, as in algebraic FETI methods [3,4].

4 REDUCTION TO BOUNDARY FREEDOMS

The expressions ofK, F and R used in the previous section pertain to all degrees of freedom of the

substructure. For many applications, notably the case of domain decomposition illustrated in Figure 5,

only the substructural boundary freedoms are involved. The interior degrees of freedom are eliminated

in some way. To illustrate the reduction scheme it is convenient to partition F, K and R as follows:

K =

Kbb KbiKib Kii

, F =

Fbb FbiFib Fi i

, R =

RbRi

, (19)

In a free-free substructure, all interior degrees of freedoms are unconstrained (that is, nodal forces are

known). Reduction to the boundary by static condensation produces the matrices

Kb = Kbb Kbi K1i i Kib, Fb = Fbb (20)

Kb is the condensed stiffness matrix, also known as a Schur complement in the mathematical literature.

Note that Kb and Fb are notdual: while the computation ofKb from K is involved, the reduction to a

boundary flexibility Fb is trivial and can be simply done by extracting appropriate rows and columns of

F. Furthermore Kb is singular while Fb is usually nonsingular and well conditioned. These aspects are

further addressed in Section 7, where efficient schemes to compute Fb from K and R are described.

Occasionally it is useful to pass directly from Fb to Kb. For example, boundary flexibilities might

be known from experimental data or a model reduction process. Denote the boundary projector by

Pb = I RbSRTb , where S = (RTb Rb)1; note that since Rb is not generally orthonormal, the kernelscaling by S must be retained. Matrices Kb and Fb = PbFbPb may be related by expressions (16) and(17), in which all matrices pertain to the boundary freedoms only. For example:

Kb = PbFb +RbSRTb )1, Fb = Pb

Kb + RbSRTb

1. (21)

The dual of the first relation only returns the projection-filtered boundary flexibility Fb = PbFbPb, andnot Fb. Thus from a boundary stiffness matrix it is generally impossible to recover a full-rank boundary

flexibility. This observation explains the superiority ofFb in model reduction processes.

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5 HANDLING A RBM-POLLUTED STIFFNESS

IfKR = 0 the stiffness matrix is said to be pollutedas regard RBMs. Physically, application of a rigidmotion to nodes produces nonzero forces. Assuming that all supports have been properly removed,

RBM pollution may arise from one or more of the following sources:

(i) Inexact arithmetic in forming K or R.

(ii) Kinematic defects in individual elements. For instance, some curved shell elements are known not

to model rotational RBMs correctly.

(iii) Kinematicdefects in assembly. The classicalexample is that of a smooth thin shell surface modeled

by faceted elements without drilling degrees of freedom. Some FEM programs delete the surface-

normal rotational freedom to preclude singularities in flat configurations, and in so doing pollute

the assembled stiffness with respect to the other two rotational RBMs.

While (i) is typically benign, (ii) and (iii) can have serious effects. Unfortunately correction at the

element and assembly level, respectively, may not be feasible because ofsoftware legacy conditions

precluding changes. One way to sanitize K post-facto is to apply congruential filtering through the

projector: K = PTKP = PKP. (22)The filtered matrix satisfies KR = 0 since PR = 0 by construction.A drawback of (22) at the substructure level is that K is generally full. This jeopardizes computations

where taking advantage of sparsity is important, such as those discussed in Section 7. Sparsity can

be recovered by doing (22) at the individual element level before assembly. This approach, however,

would not eliminate pollution from the assembly defects described under (iii) above.

The definition (7) ofF is inconvenient ifK is polluted because the spectral properties discussed in

Section 3 are lost. And so is symmetry: F = FT. A symmetric F can be obtained in two ways. Eitherthe stiffness is filtered before inversion, or the filter applied after inversion:

F = (K+ RRT)1 P, F = P(K+ RRT)1 P. (23)Both F and F are symmetric and RBM-clean. These two free-free flexibilities are generally different.

As of this writing there is not enough experience to decide on which form is best, although F is more

computationally convenient if exploitation of stiffness sparseness is important.

6 FREE-FREE FLEXIBILITY OF INDIVIDUAL ELEMENTS

The free-free flexibility of simple individual elements such as bars and beams may be computed directly

from the definition (7) as illustrated in Examples 1 and 2. For two- and three-dimensional elements it

is recommended to exploit the congruential forms outlined below.

6.1 Congruential Forms

Thefollowing expressions follow from thepseudo-inverseforms listedinSection A.1.2 of theAppendix:

K = QTSQ = F+, F = QTCQ = K+, if QQT = I and SC = I. (24)Here Q is an orthonormalized NdNF strain-displacementmatrix, S is a NdNddeformational rigiditymatrix, and C = S1 the corresponding compliance. Mathematically Pr = QTQ is the orthogonalprojector complementary to P.

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1 21 2

(a) (b)

EI

LL

u , f1 1

u , f1 1 u , f2 2

,m1 1 ,m2 2

u , f2 2

k = EA/L

Figure 6. Bar and beam elements for free-free flexibility examples.

Theelement stiffness matrixcan bemaneuvered into theform (24) through variousalgebraicor geometric

transformations without need of an explicit spectral analysis. A particularly elegant scheme consists of

orienting local axes along the principal directions of inertia of the element, as Example 3 illustrates.

For some non-simplex continuum elements generalizations of (24) that may allow analytical inversion

are

K =qi Q

T

i Si Qi , F =qi Q

T

i Ci Qi , if Qi Q

T

j = I and Si Ci = I. (25)Here the Qi matrices span q 1 mutually orthogonal subspaces, all of which are orthogonal to theRBM subspace. Elements that befit the template formulation [14], in which q = 2, may be maneuveredinto form (25) through various orthogonalization techniques.

6.2 Example 1: Bar Element

For the one-dimensional two-node bar element displayed in Figure 6(a) direct application of the defini-

tion (8) yields

K = k1 1

1 1 , R =

12

1

1 , P = 12 1 1

1 1 =

1

2kK,

F = P(K+ RRT)1 = 14k

1 1

1 1= 1

4k2K.

(26)

Here k= E A/L is the axial (equivalent spring) stiffness. It is easily verified that the result K = 4k2Falso holds for two-node bars in two and three dimensions.

Alternatively one may exploit formula (24) by following the scheme

K = QT(2k)Q, Q = 12

[1 1 ] , F = QT 12k

Q = 14k2

QT(2k)Q = 14k2

K, (27)

which gives the same result.

6.3 Example 2: Plane Beam Element

For the 2-node, 4-dof, Bernoulli-Euler prismatic plane beam element shown in Figure 6(b):

K = E IL3

12 6L 12 6L6L 4L2 6L 2L212 6L 12 6L6L 2L2 6L 4L2

, R = 12

1 L/4+L20 2/

4+L2

1 L/

4+L20 2/

4+L2

, (28)

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whence

F = L6E I(4+L2)2

2L2 L3 2L2 L3L3 24 + 12L2 + 2L4 L3 24 12L2 L42L2 L3 2L2 L3

L3

24 12L2

L4

L3

24+ 12L2

+ 2L4

. (29)

Note that entries ofF arenotdimensionally homogeneous. This is a consequence of mixing translational

and rotational nodal displacements and of normalizing the second column of R. A dimensionally

homogeneous form can be obtained by scaling the rotational freedoms by a characteristic length, as

done in the example of Figure 11.

The approach based on the congruential form (25) starts from

QT = 12

0 1 0 1

2g/L g 2g/L g

, S = E I

L

2 0

06(4+L2)

L2

(30)

in which g=

1/1+ 4/L2. It is easy to verify that QTS1Q reproduces (29).

6.4 Example 3: Plane Stress 3-Node Triangle

The simplest continuum element is the 3-node, linear displacement, plane stress triangle illustrated in

Figure 7. The element has area A, uniform thickness h, and constant 3 3 elastic modulus matrix E.Axes x and y pass through the centroid 0. With the nodal displacements arranged as

u = [ ux1 uy1 ux2 uy2 uy3 uy3 ]T (31)the free-free stiffness matrix has the well known closed form expression

K = h A BT

EB, B =1

2Ay32 0 y13 0 y21 0

0 x23 0 x31 0 x12x23 y32 x31 y13 x12 y21

, (32)

in which xi j = xi xj and yi j = yi yj . To develop the free-free flexibility, it is convenient to selectaxes ( x, y) with respect to which BTB becomes a diagonal matrix, to allow use of (26). This is done bytaking ( x, y) along the principal directions of inertia of the triangle. These are defined by the angle shown in Figure 3. The calculations proceed as follows. Defining the nodal-lumped inertias

Ixx = y221 +y232 +y231, Iyy = x212 +x223 +x213, Ixy = x12y21 +x23y32 +x13y31, (33) and related trigonometric functions are obtained as

tan2 = IxyIyy Ixx , cos2 = 1

1+ tan2 2 , sin 2 = tan2

1+ tan2 2 ,

cos2 = 12

(1+ cos2), sin2 = 12

(1 cos2),(34)

in which the angle 45 45 is chosen; if Ixx = Iyy one conventionally takes = 0. Theprincipal inertias are

Ixx = 12 (Ix x +Iyy ) + R, Iyy = 12 (Ixx +Iyy ) R, R =

14

(Iyy Ixx ) +I2xy . (35)

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-4

-3

3

2 3-2

1

0

2

3

x

y

x

y

Figure 7. Three node plane stress triangular element. Displayed triangle

has corners placed at (x, y) locations (

2,

4), (3, 1) and

(1, 3). Area A = 9/2; principal direction angle = 26.5.

Introduce the strain transformation matrix and its inverse:

Te = cos2 sin2 12 sin2sin2 cos2 1

2sin2

sin2 sin2 cos2

, T = T1e =

cos2 sin2 12 sin2sin2 cos2 1

2sin2

sin 2 sin2 cos2

, (36)

Then define

J = 14A2

Ixx 0 00 Iyy 00 0 Ixx + Iyy

, C = TTJ TTE1TJ T. (37)

Here C is a modified constitutive matrix with dimensions of compliance (because J and T are dimen-sionless). Using the formula (25) it may be verified that

F = 1Ah

BTCB. (38)

Because the work involved in computing C is minor, the work involved in forming F is not too different

from that required for thefree free stiffness (32). An alternative is touse therelation F = (AhBTEB)+ =(Ah)1B+E1(B+)T, which follows from equations (73) and (74), with B+ = (BTB)1B. This is easierto implement but lacks physical meaning.

7 FREE-FREE FLEXIBILITY OF MULTIELEMENT SUBSTRUCTURES

We nowpass to thecase of a substructure that contains multipleelements. Thesemaybe of different type;for instance combinations of beams, plates and shells. The approach of forming F(e) of each individual

element and merging is unfeasible because there is no simple flexibility assembly procedure comparable

to the Direct Stiffness Method (DSM). It is better to assemble first the free-free substructure stiffness

matrix K by the DSM. The geometric construction of the rigid body matrix R is also straightforward.

From K and R one obtains F, the computations being generally carried out in floating-point arithmetic.

We review here approaches appropriate to small and large size substructures; the crossover on present

computers being roughly 300 to 500 freedoms.

12


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7.1 Direct Calculation from Definition

The simplest calculation of the free-free flexibility F is by solving either of the linear systems

(K +RRT) F = P, or (K+ RRT)(F +RRT) = I, (39)

for F or F + RRT, respectively, following factorization of the positive-definite symmetric matrix K +RRT. In the second form RRT is subtracted to get F. Matrix K + RRT is dense since RRT generallywill fill out the matrix. Thus any sparseness initially present in K is lost. For substructures containing

less than roughly NF = 300 degrees of freedom this loss of sparsity is of little significance because theinversion of a full 300300 positive-definite symmetric matrix requires on the order of 107 operations,which is insignificant on most computers (even PCs) endowed with floating-point hardware. The two

systems (39) are equivalent ifK is RBM-clean in the sense discussed in Section 5. IfK is polluted

the first form is preferable since P acts as a filter.

Equivalent to (39) are the block-bordered symmetric systems

K RRT IR

FG = P0 , K RRT IR F+ RR

T

G = I0 , (40)

in which IR denotes the NR NR identity matrix. In exact arithmetic G = RTF = 0 and G = RT,respectively, which may be used for verification. IfK is stored in sparse form, such as a skyline format,

the coefficient matrices of (40) fit the well known arrowhead format as R generally will couple all

equations. However, forward symmetric Gauss elimination will require diagonal pivoting because K is

singular, thus hindering sparsity. Backward elimination can be completed without pivoting, but this will

fill out the K block. Hence exploitation of sparsity with (40) remains troublesome, and the additional

programming complexity tilts the balance toward the more compact arrangements (39).

7.2 Woodburying

If NF exceeds roughly 500 freedoms direct inversion becomes progressively expensive in terms of

CPU time and storage, and becomes impractical in the thousand-freedom or above range. This would

be the case, for example, when substructures arise as a result of a domain decomposition process for

task-parallel processing. Furthermore, since F is generally full, storing the complete flexibility would

demand significant storage resources. Fortunately, as noted in Section 4, more often than not only a

boundary subsetofF needs to be computed, which reduces the need for storage. It is therefore of

interest to develop computational techniques that take advantage of:

(i) The sparseness ofK, in the sense that pivoting on the K block is precluded.

(ii) The low rank ofR.

(iii) The need for only a boundary subset ofF.

If only RBMs are allowed in R this matrix has at most rank 6. It is therefore tempting to consider

using the Woodbury inverse-update formula for (K + RRT)1. The standard formula cannot be used,however, because K is singular. A work-around is developed in the Appendix: the formula is applied

to K+HHT, where H is a NFNR matrix such that HHT is a diagonal matrix of rankNR that rendersK+ HHT positive definite while preserving its sparsity and makes pivoting unnecessary.The necessary theory is worked out in Section A.3 of the Appendix, and only the relevant results are

transcribed here. In those equations replace A by K, B by F, n by NF and kby NR , while keeping the

13


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same notation for other matrices and vectors. The first of (39) is replaced by

(K +HHTHHT+ RRT) F = P. (41)

The equivalent block-symmetric form is

K +HHT R H

RT IR 0HT 0 IR

F

GRGH

=

P

0

0

. (42)

Since K+HHT is positive definite no pivoting is necessary while processing that block, which containsthe bulk of the equations. The explicit matrix borderings of (42) can be avoided by doing two linear

blocksolves followed by a matrix combination:

(K+ HHT) X = I, (K +HHT) YR = R,F = P X P = X YRRT RYTR + RRTYRRT.

(43)

In the first stage, K + HHT is factored by a sparse symmetric solver and NF+ NR right hand sides,namely the columns ofI and R, solved for. The second (combination) stage involves dyadic matrix

multiplications, which may be resequenced as Z = X P = X YRRT followed by F = P Z =Z RRTZ. This reorganization is useful when computing a subset Fbb ofF, as further discussed inSection 7.5.

7.3 Constructing H

To close the algorithm specification, it is necessary to state how H is constructed. The scheme (43) is

valid for an arbitrary H matrix such that HTR has full rank. To preserve the sparsity ofK, however,

HHT is required to be diagonal. That requirement is easily handled by choosing the columns of H to

be elementary vectors hj . This vector has all zero entries except for its jth entry, which is set to sj > 0.

Obviously hj hTj is the null matrix except for the j

th diagonal entry, which is s2j . Since this is added to

Kj j , s2

j can be viewed as the stiffness of a penalty spring added to the jth freedom. The goal is to insert

NR springs to suppress the NR rigid body modes. Two insertion strategies may be followed:

1. Prescribed insertion. Penalty spring stiffnesses and their locations are picked beforehand from

inspection or separate analysis of the substructure.

2. Adaptive insertion. The NR degrees of freedom are inserted on the fly as a byproduct of the

factorization ofK, as described below. This strategy has the advantage of providing consistency

crosschecks, and is the one selected here.

This kind of adaptive strategy was originally proposed for the stable traversal of critical points ingeometrically nonlinear analysis [15,16]. Although there are procedural differences (in critical point

traversal usually one spring is sufficient) the underlying idea is the same.

The following tests are patterned after the skyline solver implementation described in [17], but are

applicable with minor changes to most direct solvers. We assume that the sparse factorization program

carries out the symmetric LDLT decomposition

K +HHT = LDLT, (44)

14


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where D is diagonal and L unit lower triangular. As preprocessing step the factoring program computes

the Euclidean row lengths j ofK and their running maxima mj , which are saved in a work array:

j =

NFi=1 K

2i j

1/2

, mj = maxji=1i , j = 1, . . . NF. (45)

Initially H = 0. Suppose that the factorization (44) has reached the j th column and row. Entry dj ofD(known as a diagonal pivot) is computed as last substep. The following singularity test is performed:

|dj | Cd mj = Ctol mj . (46)

Here is a machine tolerance (the smallest number for which 1+ > 1 in the floating-point arithmeticused) and Cd is a loosen up coefficient typically in the range 10 to 100. If (46) holds, a penalty spring

with s2j = Cs mj , where Cs is a coefficient of order 102 or 103, is added to dj so effectively dj s2j .The penalty spring count is incremented by one and the factorization continued.

IfK is found tobe NR times singular as per (46), NR springs are inserted. On exit, theadaptive procedure

has effectively changed K by the diagonal matrix HHT of rankNR defined as

DH = HHT =NRi=1

hj hTj , where i

th entry ofhj = sj > 0 ifi = j , else 0. (47)

H is the NFNR rectangular matrix obtained by stacking the hj s as columns. For example, supposethat NF = 6, NR = 2 and that freedoms j = 3 and j = 6 receive penalty springs. Then

h3 =

0

0

s30

0

0

, h6 =

0

0

00

0

s6

, H = [ h3 h6 ] =

0 0

0 0

s3 00 0

0 0

0 s6

, HHT =

0 0 0 0 0 0

0 0 0 0 0 0

0 0 s23 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 s26

. (48)

It is important to note that H is used in the theoretical developments for convenience but is never formed

explicitly.

The foregoing scheme is called an exact penalty methodin the sense that in exact arithmetic the penalty

spring values have no effect on the final result for F, as long as they are positive nonzero. In inexact

floating-point arithmetic it is wise to constrain the sj to a certain range to minimize rounding errors and

cancellations, but within that range the results remain insensitive to the value. The recommended range

s2j = 100mj to 1000mj fulfills that objective.On factorization exit it is necessary to verify whether the penalty spring count is NR . This a posteriori

consistency check is discussed in Section 7.9.

7.4 Tutorial Example: Three Springs in Series

This simple example is worked out in detail to display the equivalence between the direct and penalty-

spring flexibility computations. It consists of three springs of stiffnesses k1, k2 and k3 attached in series,

15


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1 2 3 4(1) (2) (3)

k1 k2 k3

x

u2u1 u3 u4

Figure 8. Three springs in series.

as shown in Figure 8. The springs can only move longitudinally and consequently the substructure has

four degrees of freedom: u1 through u4. The stiffness, rigid-body and projector matrices are

K =

k1 k2 0 0k2 k1 + k2 k2 0

0 k2 k2 + k3 k30 0 k3 k3

, R = 12

1

1

1

1

, P = 14

3 1 1 11 3 1 11 1 3 11 1 1 3

(49)

Ifk1=

k2=

k3=

1, a direct calculation gives

F = P(K+ RRT)1 = (K+ RRT)1P = 18

7 1 3 51 3 1 3

3 1 3 15 3 1 7

. (50)

The calculation is now repeated with the method of penalty springs. Because NR = 1 one spring hasto be injected. The adaptive collocation method will insert s2 at freedom u4 because the symmetric

forward factorization ofK will get a zero or tiny pivot there. Thus

H

= 0

0

0s

, KH = K+ HHT

= 1 1 0 01 2 1 00 1 2 10 0 1 1+ s2

. (51)We will keep s arbitrary to study its propagation in the calculation sequence. Solving the systems

KHX = I and KHYR = R we obtain

X = 14

3+ 1/s2 2+ 1/s2 1+ 1/s2 1/s22+ 1/s2 2+ 1/s2 1+ 1/s2 1/s21+ 1/s2 1+ 1/s2 1+ 1/s2 1/s2

1/s2 1/s2 1/s2 1/s2

, YR = 12

6+ 4/s25+ 4/s23+ 4/s2

4/s2

, (52)

and combining:

F = X YTRRTRYTR R RTYRRT = 18 7 1 3 51 3 1 33 1 3 15 3 1 7

. (53)As expected the effect of a nonzero s cancels out in F because of the exact arithmetic. Numerical

computations in double-precision floating-point show that F is obtained with full (16-place) accuracy

ifs > 1 (but not so large as to cause overflow). Ifs < 1, the accuracy in X drops because K + HHTapproaches a singular matrix, since its condition number is C(KH) 13.6/s2 for s


19/33

7.5 Computing a Subset of F

As noted in Section 4, for many applications of the free-free flexibility it is only necessary to evaluate

a submatrix Fb Fbb of F that pertains to Nb < NF freedoms at boundary nodes. The penaltyspring procedure can be adjusted to cut down on both computations and storage. The changes are best

illustrated through a modification of the previous example. Suppose that only the end freedoms (u1, f1)

and (u4, f4) are of interest for Fbb. The construction ofK, H and KH = K+HHT proceeds as beforeand the solution for YR does not change. The linear system KH X = I may be reduced to KH Xb = Ib,in which the 42 matrices Xb and Ib hold columns 1 and 4 ofX and I, respectively. A trimmed versionRb of the R matrix that keeps only rows 1 and 4 is formed. For this example:

Xb = 14

3+ 1/s2 1/s22+ 1/s2 1/s21+ 1/s2 1/s2

1/s2 1/s2

, YR = 12

6+ 4/s25+ 4/s23+ 4/s2

4/s2

, R Rb = 12

1

1

. (54)

The NF

Nb matrix Zb

=Xb

YRR

Tb is computed, and a trimmed version Zbb formed by keeping

only rows 1 and 4:

Zb = Xb YRRTb = 14

6 63 51 30 0

Zbb = 14

6 60 0

. (55)

Combining:

Fbb = Zbb Rb RTZb = 18

7 55 7

. (56)

The projected boundary flexibility is Fb = PbFbbPb = (3/4) 1 11 1, in which Pb = I2 Rb(R

Tb Rb)

1RTb . This is the pseudo inverse of the condensed stiffness Kb = (1/3)

1 11 1

.

For this simple problem these gyrations are hardly worth the trouble. But they make sense for larger

systems. For example, a substructure made of a 50 50 plane stress quadrilateral mesh contains 400boundary and 4802 internal freedoms, respectively. Hence F would be 5202 5202, requiring 210MBfor storage in double precision as a full matrix. But the 400 400 matrix Fbb would need only 1.3MB.

7.6 A Model Reduction Example

This example illustrates the application ofF to a model reduction process that retains only selected

boundary freedoms. It can also be used as a benchmark problem to test implementations of the exactpenalty spring method. Figure 9(a) shows a 16-element, free-free substructure built with 4-node square

plane stress elements with two freedoms per node. The mesh has 25 nodes and NF = 50 degrees offreedom, and three independent rigid body modes (NR = 3). The plate dimensions and thickness areshown in the figure. The material is isotropic with E= 160 and = 1/3 except for element 6, whichis taken to have either E(6) = 0 (hole) or E(6) = 160108 (near-rigid inclusion). The reduced modelretains only the 10 horizontal displacements and forces at the boundary nodes shown in Figure 9(b).

(There are 32 boundary freedoms, but only 10 are kept here to keep Fbb printable.)

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(a) (b)

17

18

19

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

20

21

22

23

24

25

1

2

3

4

5

21

22

23

24

25

fx1

fx2

fx3

fx4

fx5

fx21

fx22

fx23

fx24

fx25

32

32

(1) (5) (9) (13)

(2) (6) (10) (14)

(3) (7) (11) (15)

(4) (8) (12) (16)

x

y

z

thickness h = 1/100,

E= 0 or 160 10 , = 1/3

thickness h = 1/100,

E= 160, = 1/38

modelreduction

Figure 9. Application ofF to model reduction: (a) a 25-node, 50-DOF plane-stress substructure;

(b) a reduced model that retains only 10 boundary freedoms. The shaded element (6)

models either a hole (E(6) = 0) or a near-rigid inclusion (E(6) = 160 108).

Although the problem size is too small for sparse computations to be advantageous, it facilitates the

visualization of matrix configurations as depicted in Figure 10.

With the {x, y} axes placed as shown, the 50 3 orthonormalized RBM matrix R is

RT = 110

2 0 2 0 2 0 2 0 2 0 2 0 . . . 2 0

0 2 0 2 0 2 0 2 0 2 0 2 . . . 0 2

2 2 1 2 0 2 1 2 2 2 2 1 . . . 2 2

(57)

The isoparametric element stiffness matrix for

=1/3, h

=1/100 and variable E is

K(e) = E1600

8 3 5 0 4 3 1 03 8 0 1 3 4 0 5

5 0 8 3 1 0 4 30 1 3 8 0 5 3 4

4 3 1 0 8 3 5 03 4 0 5 3 8 0 1

1 0 4 3 5 0 8 30 5 3 4 0 1 3 8

, (58)

from which the 50 50 free-free stiffness matrix K is easily assembled by the DSM.For the hole case, with E(6) = 0 and E= 160 elsewhere, the diagonal entries ofK are

diag(K) = 15

[4 4 8 8 8 8 8 8 4 4 8 8 12 12 12 12 16 16 8 8 . . . 8 8 4 4 ] . (59)

The rowlength maxima, listed to 3 places, are

mj = [ 1.11 1.11 2.02 2.02 2.02 2.02 2.02 2.02 2.02 2.022.02 2.02 2.81 2.81 2.81 2.81 3.65 3.65 . . . 3.65 ]

(60)

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1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

N = 50F N =3R

K + H H I RT

Figure 10. Leftmost diagram shows the configuration ofK

+HHT for the substructure of Figure 9.

Each square represents a 2 2 block. White squares mark zero entries. The rightmostdiagrams depict the configuration of the right-hand side matrices I, and R.

Border numbers above I are node numbers. Shaded columns in I mark the

10 columns ofI to be processed. Black circles mark the unit diagonal elements at which

processing starts. Three penalty springs suppressing the RBMs are inserted at freedoms

47, 49 and 50. These are marked with black squares in the diagonal ofK.

On factoring K = LDLT in double-precision floating-point the first 47 diagonal entries ofD, listed to4 decimal places, are

diag(D)[1:47] = [ 0.8000 0.6875 1.5854 1.2368 . . . 0.1851 0.7415 1.066 1014 ] (61)

With Ctol = 1013 used in the test (46), the first singularity is detected at j = 47 because d47 =1.0661014 < 10133.65. A penalty spring s247 = 100 3.65 = 365 is inserted, whence d47becomes s247 to 14 places. The factorization continues with d48 = 0.6084 and d49 = 9.3261015. Thesecond spring is inserted at j = 49 changing d49 also to 365. The third singularity is detected at thelast entry: d50 = 9.9921016. The third spring goes at j = 50 and d50 becomes 365. On completingthe remaining steps in (43) the following reduced flexibility, listed to 4 decimal places, is obtained:

Fbb =

2.0510 0.0587 0.2545 0.2086 0.0018 0.4334 0.2064 0.1644 0.1145 0.02951.0558 0.1168 0.1083 0.1746 0.1971 0.1462 0.1680 0.1518 0.0843

1.0647 0.1420 0.2225 0.2334 0.1676 0.1172 0.0846 0.08940.9642 0.0260 0.1486 0.1274 0.0843 0.0808 0.1920

1.9373 0.0950 0.0489 0.1240 0.2237 0.50662.0534 0.0313 0.2078 0.1960 0.0556

0.9341 0.0831 0.0797 0.17310.9442 0.0934 0.1841

0.9313 0.0080

symmetric 1.9614

(62)

Its eigenvalues are: [ 2.6335 2.4016 2.0182 1.4903 1.4570 1.0796 0.9497 0.7887 0.7717 0.3071].

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Comparison with the exact full F obtained in rational arithmetic with Mathematica shows that the

double precision floating-point computation sequence delivers 15 places of accuracy.

For the near-rigid intrusion case, in which E(6) = 160 108 and E = 160 elsewhere, the diagonalentries ofK are as in (59) except at nodes 7, 8, 12 and 13 (j = 13, 14, 15, 16, 23, 24, 25, 26) wherethey jump to 8000002.4. The rowlength maxima mj start as in (60), but jump to 1.1136

107 at j

=13

and stay at that value through j = 50. The factorization proceeds without incident until reachingd47 = 2.19108, which flags a singularity since 2.19108 < 10131.1136107. A penalty springofs247 = 1001.1136107 = 1.1136 109 is added there. The next tiny pivot is d49 = 1.028108,immediately followed by d50 = 1.31751012. The same penalty spring values are inserted at j = 49and j = 50. Proceeding with the remaining steps the following reduced flexibility, listed to 4 decimalplaces, is obtained:

Fbb =

1.8371 0.0633 0.1338 0.1101 0.0138 0.5129 0.2392 0.1377 0.0480 0.11810.7467 0.0018 0.0542 0.1238 0.2237 0.0675 0.0323 0.0559 0.0674

0.7546 0.0068 0.1519 0.1007 0.0346 0.0250 0.0799 0.17620.8500 0.0041 0.0321 0.0725 0.0898 0.1230 0.2503

1.9126 0.0859 0.0832 0.1618 0.2379 0.48861.8441 0.0470 0.1767 0.1204 0.01830.8622 0.0396 0.0776 0.1281

0.8632 0.0398 0.18300.8729 0.0270

symmetric 1.9030

(63)

Its eigenvalues are [ 2.5252 2.2875 1.8236 1.2890 1.0172 0.9076 0.8070 0.7698 0.7294 0.2902 ].

Comparison with the exact F obtained in rational arithmetic shows that the double precision floating-

point computation delivers on average 11 places of accuracy. Hence the effect of the poorly scaled

stiffness has been to lose 4 decimal places with respect to the well scaled (hole) case.

Although the full stiffness K and flexibility F of the two cases have entries that differ by eight orders ofmagnitude, the reduced-model flexibilities are of the same order. Furthermore, observe that both (62)

and (63) are diagonal dominant and very well conditioned. This ability of boundary flexibilities to filter

out interior details is a key factor in the success of FETI-type iterative solvers [1,2,4,3].

7.7 A Substructure with Internal Mechanisms

The last example illustrates how to handle a substructure that exhibits non-RBM zero energy modes.

Symbolic analysis is used to make the result valid for a wide range of property data and hence more

usefulas validationbenchmark. Threeidenticalprismaticplanebeam-column members areconnectedas

shown inFigure11(a). Thehinge at node 4 permits members to rotateindependently: (1)4 = (2)4 = (3)4 ,

while enforcing equal translations. The system has the 14 degrees of freedom defined in Figure 11(a),

which are ordered as

uT = [ ux1 uy1 L1 ux2 uy2 L2 ux3 uy3 L3 ux4 uy4 L(1)4 L(2)4 L(3)4 ] (64)

The rotational freedoms are scaled by L to get dimensionally homogeneous results. The three members

have the same length L, elastic modulus E, cross section area A and moment of inertia I= Izz . Forconvenience in subsequent symbolic expressions, we set I = r2A = 2AL2, where r is the cross-section radius of gyration and = r/L the bending slenderness ratio. The assembled free-free stiffness

20


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1

3

2

4

1

2

3

4(1)

4(3)

(2)4

(1)

(3)

(2)uy1

uy2

ux2

uy3

uy4

ux4

ux3

ux1

E, A, I andL same for

the three elements

(a) (b)

NBM #3

NBM #4

NBM #5

Hinge

L

60o

60o

x

y

Figure 11. Three identical plane beam-columns hinged at node 4. Figure (b) depicts3 null basis modes (NBM) used to start the construction of the 5 14 nullspace basis matrix (66); NBMs #1 and #2 are the translational RBMs.

is

K = E AL

1 0 0 0 0 0 0 0 0 1 0 0 0 00 122 62 0 0 0 0 0 0 0 122 62 0 00 62 42 0 0 0 0 0 0 0 62 22 0 00 0 0 2 1 5 0 0 0 2 1 0 5 00 0 0 1 3 32 0 0 0 1 3 0 32 00 0 0 5 32 42 0 0 0 5 32 0 22 00 0 0 0 0 0 2 1 5 2 1 0 0 50 0 0 0 0 0 1 3 32 1 3 0 0 320 0 0 0 0 0 5 32 42 5 32 0 0 221 0 0 2 1 5 2 1 5 4 0 0 5 50 122 62 1 3 32 1 3 32 0 4 62 32 320 6

2

22

0 0 0 0 0 0 0 62

42

0 00 0 0 5 32 22 0 0 0 5 32 0 42 00 0 0 0 0 0 5 32 22 5 32 0 0 42

,

(65)

in which 1 = 14

3(1 122), 2 = 14 + 92, 3 = 34 + 32, 4 = 32 + 182 and 5 = 3

32.

The 514 null space basis will be also denoted by R. This matrix contains the 3 rigid-body modes plustwo mechanisms, and can be constructed directly by geometric inspection without having to analyze

K. As initial basis we select the two translational RBMs along x and y, plus the three individual beam

21


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rotations depicted in Figure 11(b). Applying Gram-Schmid yields the orthonormal basis

RT

=

12

0 0 12

0 0 12

0 0 12

0 0 0 0

0 12

0 0 12

0 0 12

0 0 12

0 0 0

0

31 41 0 1 0 0 1 0 0 1 41 0 0

2 23 3 32 83 4 2 25 0 2 25 3 4 067 46 26 57 76 26 177 176 466 67 66 26 26 8

, (66)

in which 1 = 1/(2

11), 2 = 12

11/161, 3 = 2/

5313, 4 = 4

11/483, 5 =

3/1771,

6 = 1/(6

161), 7 = 1/(2

483) and 8 =

23/63. A symbolic calculation with Mathematica

using the method of penalty springs delivers the following boundary flexibility associated with the nine

degrees of freedom at nodes 1, 2 and 3:

Fbb =L

5292E A2

1 0 0 6 9 10 6 9 100 2 7 28 11 12 28 11 120 7 3 8 13 14 8 13 146 28 8 4 15 16 17 18 19

9 11 13 15 5 20 18 21 2210 12 14 16 20 3 19 22 146 28 8 17 18 19 4 15 169 11 13 18 21 22 15 5 2010 12 14 19 22 14 16 20 3

, (67)

in which 1 = 24 + 29162, 2 = 132 + 2882, 3 = 1356 + 722, 4 = 105(1 + 92), 5 =51 + 22592, 6 = 6(4 + 1712), 7 = 66 + 1442, 8 = 3

3 (1 362), 9 = 2

3 (8 2252),

10 = 2

3 (5 + 722), 11 = 46 + 3602, 12 = 16 722, 13 = 23 + 1802, 14 = 8 362,

15 =9

3 (3

732), 16 =

3

3 (11+

242), 17 =

3(19+

92), 18 =

3 (5

1172),

19 = 3 (13+ 362), 20 = 33+ 722, 21 = 13+ 13592 and 22 = 7+ 2522. Symbolic penaltysprings are inserted at K-factorization pivots 10, 11, 12, 13 and 14, which become exact zeroes in the

exact arithmetic used by Mathematica.

Matrix (67) has full rank of 9 if > 0. If , which for fixed A and L means that the 3 membersbecome infinitely stiff in bending, Fbb approaches a finite matrix of rank 3 whereas K .

7.8 Computational Costs

If a sparse skyline solver is used for the factor and blocksolve stages, the amount of arithmetic work

required in the penalty spring method may be estimated as follows. Denote by b the root-mean-square

bandwidth ofK, n

=NF the order ofK and k

=NR the RBM count. The cost in floating-point

operation units to get the full F is approximately

Ctot = CF+ CS+ CC, CF = 12 nb2, CS= 2 n b( 12 n + k), CS= 3n2k, (68)

where CF, CS, and CC denote the costs of KH-factorization, blocksolve and matrix combination,

respectively. For CS the special form of the RHS in block solving AHX = I has been accounted for.The storage requirement is approximately S= nb+ 1

2n2+2nk double precision floating-point (8-byte)

locations.

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Table 1 - Computational Costs and Storage Demands for Sample 2D/3D Regular Meshes

( Cs in billions offloating-point operation units, storage in gigabytes)

Case N n=

NF

nb

CF

CS

CC

Ctot

Sin GB

2D F 50 5000 5000 0.03 2.50 0.22 2.75 0.102D F 100 20000 20000 0.40 80.02 3.60 84.02 1.632D F 200 80000 80000 6.40 2560.20 57.60 2624.20 25.862D Fbb 50 5000 400 0.03 0.39 0.02 0.44 0.022D Fbb 100 20000 800 0.40 6.30 0.14 6.84 0.162D Fbb 200 80000 1600 6.40 101.57 1.15 109.12 1.293D F 10 3000 3000 0.2 2.7 0.2 3.1 0.043D F 20 24000 24000 17.3 691.6 10.4 719.2 2.543D

F 40 192000 192000 2211.8 176958.4 663.6 179833.8 154.85

3D Fbb 10 3000 1800 0.2 2.3 0.1 2.6 0.063D Fbb 20 24000 7200 17.3 352.9 3.1 373.2 1.823D Fbb 40 192000 28800 2211.8 49113.9 99.6 51425.3 54.95

If only a subset Fbb or order nb < n is required, then again Ctot = CF+ CS+ CC, where now

CF = 12 nb2, CS= 2 n b

(1 nb2n

)nb + k

, CS= 3 n nb k. (69)

Thus CF stays the same whereas CS and CC drop. The storage required is approximately S= nb +nnb + 12 n2b + 2nk double precision floating-point (8-byte) locations.To provide a more concrete cost picture, two regular mesh configurations are chosen and results posted

in Table 1. The case labeled 2D pertains to a regular N N two-dimensional plane stress meshof 4-node quadrilaterals with two freedoms per node, n = NF 2N2, b 2N, k= NR = 3 andnb = 8N. Table 1 lists costs (in billions of operation units) for N= 50, 100 and 200 to compute thefull F and the boundary-reduced Fbb . The leading term in Ctot is 8N

5 for F and 68N4 for Fbb . In both

cases the blocksolve cost CS is dominant. On a one-gigaflop (1GF) CPU, typical of present high-end

offerings by Intel and AMD, the estimate runtime in seconds will be that shown on Table 1 for Ctot; for

instance 109 seconds to get the Fbb of a 200 200 mesh.The case labeled 3D pertains to a regular N

N

N three-dimensional mesh of 8-node bricks with

three freedoms per node, n = NF 3N3, b 3N2, k= NR = 6 and nb 18N2. The leading term inCtot is 27N

8 for F and 338N7 for Fbb . Table 1 lists data for N= 10, 20 and 40. The total cost is againstrongly dominated by the blocksolve. On a 1GF processor, the computation ofFbb for a 20 20 20mesh would take 373 seconds. Note that the ratio of full to boundary-reduced costs is not so significant

as in the 2D case because a larger percentage of nodes are on the boundary; for example that ratio is

only about 2 for N= 20.

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Several conclusions emerge from the cost analysis:

1. The factorization cost is comparatively insignificant for fine meshes in 2D and 3D. Multiple

refactorization passes, for example to test the effect of different singularity tolerances on the

penalty spring count, would have minor impact on the total cost.

2. Improved computational efficiency in the blocksolve, for example using assembly language to

squeeze high performance out of superscalar or pipeline processors, would have a more immediate

payoff than improvements in the sparse factorization. For the same reason, complicated sparse

solvers with high indexing overhead should be avoided.

3. The estimated CS may be used as a guide in model decomposition to attain load balance in parallel

computations where one or more substructures are assigned to each processor. The solution cost

will also dominate the parallel recovery of internal states if the factorization ofKH is saved in each

processor.

It should be noted that the cost of forming the projected boundary flexibility Fb = Pb Fbb Pb describedin Section 4 is not included in these estimates. This may be viewed as a postprocessing step required

for some applications.

7.9 Implementation Considerations

The determination of rank in floating-point arithmetic is a delicate treshold problem [10,18]. Ac-

cordingly, when processing general substructures the penalty spring method should be complemented

with appropriate safeguards to achieve robustness. A software configuration that implements two con-

sistency tests is diagramed in Figure 12.

Upon forming (or procuring) K and R, an obvious check is whether KR = 0 within floating-pointtolerance. Failure flags serious inconsistencies; for example K may come from a black box source

such as a commercial FEM code and be RBM polluted, or the substructure geometric data is flawed.

An immediate error exit is indicated.

The second test checks whether the penalty spring count Ns returned by the sparse factor agrees with

the column dimension NR ofR. If so, processing continues with the blocksolve and combination steps

to return either F or Fbb. Handling discrepancies is tricky because many error sources, model-based

or computational, are possible. A spring count exceeding NR may be due to (i) spurious kinematic

mechanisms, (ii) highly ill-conditioned K, or (iii) overly loose tolerance in the singularity test (46). A

count less than NR may be due to (iv) overstrict singularity test tolerance, (v) RBM polluted stiffness,

or (vi) physical mechanisms (such as initial stress effects in geometrically nonlinear analysis) that

eliminate some RBMs. (The last two sources, however, should be caught by the prior KR = 0? test.)If a count discrepancy occurs, it is recommended to proceed to an in-depth diagnosis of the null space

ofK, as flowcharted in Figure 12. This may involve a SVD analysis [19] or a geometric analysis

complemented by the SVD [20]. The interested reader should consult those references for proceduraldetails. On return from this step a decision must be made to proceed, or to take an error exit requiring

further instructions from the user. Two scenarios are:

1. The null-space analyzer confirms the initial R. The factorization singularity tolerance should be

then adjusted until the correct number of springs is returned (discrepancy one way or the other is

not acceptable and will result in an incorrect flexibility). As noted in the cost study, the expense of

multiple refactorizations is negligible for fine 2D and 3D meshes. If agreement cannot be achieved

K is likely to be extremely ill-conditioned and/or poorly scaled, and the model substructuring

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Yes

Yes

No

NoFull F

requestedReduced Frequested

b

ELEMENT

LIBRARY

SUBSTRUCTURE

GEOMETRY and

CONNECTIVITY

TABLES

FOR THIS ANALYSIS SEE

REFS. CITED in TEXT

SPARSE MATRIX

LIBRARY

Sparserowsum

Sparsefactor

Sparseblocksolve

Combineto get F

R

Solve forX & Y

StiffnessAssembler

Form R of substructure

from geometry andorthonormalize

Geometricanalyzer

SVDlibrary

Successful exit

Entry

D,L, spring count

K

K

K

R

RX,Y

D,L,R,I

mj

RbX ,Yb

R

Solve for X& Y , trim to

X & YRbbb

b

Fatalerrors

bbF

Combineto get Fbb

F

Error exits:polluted K,

incorrect R, orillegal mechanisms

Form K of substructure(unless supplied on entry)

Test whether penaltyspring count agrees withcolumn dimension ofR

Confirmedor updated R

In-depth substructurenull space analysis by

SVD and geometry

Factor K = LDL injectingpenalty springs. Record

number and location

TTest whether K R = 0

Figure 12. Schematics of the free-free flexibility analysis of a substructure. For the

in-depth null space analysis, which is not treated here, see [19] and [20].

process should be revised.

2. Additional kinematic mechanisms are found. If these are disallowed an error exit is taken. Ifallowed and the spring count agrees with the null space dimension, the geometric R is replaced

by the updated null space basis and the blocksolver path taken. If acceptable but the spring count

disagrees, the factorization path is retraced with adjusted tolerances as above.

The ultimate solution to fatal error conditions is revision of the model decomposition process and impo-

sition of element quality control (in particular, to eliminate the pollution errors discussed in Section 5)

until all substructures behave as expected. For this reason, it pays to be conservative in handling error

conditions at this level.

A questionmayberaisedas towhether an in-depthnull spaceanalysisof thesubstructural stiffness should

be always done on entry. The philosophy of the present implementation is that such analysis serves

primarily to catch model or substructuring flaws. Once those are fixed, it is not only superfluous but maycontaminate the geometric R with roundoff noise. The analysis of complex structural configurations

typically requires hundreds or thousands of production runs for design modifications during which the

substructuring is kept unchanged. It is better to keep the null space analysis as a background tool for

verification and troubleshooting of the initial passes.

A more difficult decision is: should non-RBM substructural kinematic mechanisms be forbidden? Our

recommendation is to allow only those that possess physical significance; for example vehicle steering

and suspension linkages, or control surface motions in aircraft. These mechanisms can usually be

25


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defined geometrically by inspection, as in the example of Figure 11, again making a detailed null space

analysis ofK superfluous.

8 CONCLUSIONS

The introduction of the free-free flexibility closes a duality gap with the stiffness method. This paper

treats the subject at the level of individual substructures that arise from a model decomposition process.The major new contributions presented here are:

1. An exact penalty method to compute F or a subset thereof, given K and R, for arbitrary sub-

structures. The method preserves the sparseness structure ofK and can be implemented in the

framework of existing symmetric sparse linear solvers. To increase robustness it is recommended

tocomplement the basic algorithm witha null space analyzer to handle difficult or borderlinecases.

2. A general expression for F, derived through the Woodbury formula, that involves a rank-

regularization matrix H.

3. Congruential transformation methods for symbolic derivation of the free-free flexibility of indi-

vidual elements.

4. Dual and nondual relations that connect boundary-reduced stiffness and flexibility matrices, which

are expected to be useful in model reduction and system identification.

Further refinements in null space analysis of substructures and subdomains can be expected as interest

grows in multilevel parallel solution of extremely large FEM models, containing tens or hundreds

of millions of freedoms. The solution framework may involve boundary flexibilities connected by

displacement frames and should be able to accommodate nonmatching meshes as well as nonlocal

(inter-subdomain) multipoint constraints.

Acknowledgements

The present work has been supported by Lawrence Livermore National Laboratory under the Scalable Algorithms

for Massively Parallel Computations (ASCI level-II) contract B347880, by Sandia National Laboratory under theAccelerated Strategic Computational Initiative (ASCI) contract AS-5666, and by the National Science Foundation

under award ECS-9725504.

References

[1] C. Farhat and F.-X. Roux, A method offinite element tearing and interconnecting and its parallel solution

algorithm, Int. J. Numer. Meth. Engrg., 32, 12051227, 1991.

[2] C. Farhat and F.-X. Roux, Implicit parallel processing in structural mechanics, Computational Mechanics

Advances, 2, 1124, 1994.

[3] K. C. Park, M. R. JustinoF. and C. A. Felippa, An algebraically partitioned FETI method for parallel structural

analysis: algorithm description, Int. J. Numer. Meth. Engrg., 40, 27172737, 1997.

[4] M. R. Justino F., K. C. Park and C. A. Felippa, An algebraically partitioned FETI method for parallel structural

analysis: performance evaluation, Int. J. Numer. Meth. Engrg., 40, 27392758, 1997.

[5] U. Gumaste, K. C. Park and K. F. Alvin, A family of implicit partitioned time integration algorithms for

parallel analysis of heterogeneous structural systems, Comput. Mech., 24, 463475, 2000.

[6] K. C. Park and K. F. Alvin, Extraction of substructural flexibility from measured global modes and mode

shapes, Proc. 1996 AIAA SDM Conference, Paper No. AIAA 96-1297, Salt Lake City, Utah, April 1996,

also AIAA J., 35, 11871194, 1997.

[7] K. C. Park, G. W. Reich and K. F. Alvin, Damage detection using localized flexibilities, in Structural Health

Monitoring, Current Status and Perspectives, ed. by F.-K. Chang, Technomic Pub., 125139, 1997.

26


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[8] K. C. Park and G. W. Reich, A theory for strain-based structural system identification, Proc. 9th International

Conference on Adaptive Structures and Technologies, 1416 October 1998, Cambridge, MA.

[9] K. C. Park and C. A. Felippa, A flexibility-based inverse algorithm for identification of structural joint

properties, Proc. ASME Symposium on Computational Methods on Inverse Problems, Anaheim, CA, 1520

Nov 1998.

[10] G. W. Stewart and J. Sun, Matrix Perturbation Theory, Academic Press, New York, 1990.[11] C. A. Felippa, K. C. Park and M. R. Justino F., The construction of free-free flexibility matrices as generalized

stiffness inverses, Computers & Structures, 88, 411418, 1998.

[12] C. A. Felippa and K. C. Park, A direct flexibility method, Comp. Meths. Appl. Mech. Engrg., 149, 319337,

1997.

[13] S. L. Bott and R. J. Duffin, On the algebra of networks, Trans. Amer. Math. Soc., 74, 99109, 1953.

[14] C. A. Felippa, Recent advances in finite element templates, Chapter 4 in Computational Mechanics for the

Twenty-First Century, ed. by B.H.V. Topping, Saxe-Coburn Publications, Edinburgh, 7198, 2000.

[15] C. A. Felippa, Penalty spring stabilization of singular jacobians, J. Appl. Mech., 54, 17301733, 1987.

[16] C. A. Felippa, Traversing critical points by penalty springs, Proceedings of NUMETA87 Conference,

Swansea, Wales, M. Nijhoff Pubs, Dordrecht, Holland, July 1987.

[17] C. A. Felippa, Solution of equations with skylinestored symmetric coefficient matrix, Computers & Struc-tures, 5, 1325, 1975.

[18] C. L. Lawson and R. J. Hanson, Solving Least Squares Problems, Prentice Hall, 1974.

[19] C. Farhat and M. Geradin, On the general solution by a direct method of a large scale singular system of

equations: application to the analysis offloating structures, Int. J. Numer. Meth. Engrg., 41, 675696, 1998.

[20] M. Papadrakakis and Y. Fragakis, An integrated geometric-algebraic approach for solving semi-definite

problems in structural mechanics, Comp. Meths. Appl. Mech. Engrg., 190, 65136532, 2001.

[21] C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and its Applications, Wiley, New York, 1971.

[22] R. E. Cline, Note on the generalized inverse of the product of matrices, SIAM Review, 6, 5758, 1964.

[23] S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Dover, NY, 1991.

[24] W. W. Hager, Updating the inverse of a matrix, SIAM Review, 31, 221239, 1989.

[25] H. V. Henderson and S. R. Searle, On deriving the inverse of a sum of matrices, SIAM Review, 23, 5360,

1981.

[26] M. Woodbury, Inverting modified matrices, Memorandum Report 42, Statistical Research Group, Princeton

University, Princeton, NJ, 1950.

[27] A. Householder, The Theory of Matrices in Numerical Analysis, Blaisdell, 1964.

[28] J. Sherman and W. J. Morrison, Adjustment of an inverse matrix corresponding to changes in the elements of

a given column or a given row of the original matrix, Ann. Math. Statist., 20, 621, 1949; also Adjustments of

an inverse matrix corresponding to a change in one element of a given matrix, Ann. Math. Statist., 21, 124,

1950.

[29] J. A. Fill and D. E. Fishkind, The Moore-Penrose generalized inverse for sum of matrices, SIAM J. Matrix

Anal. Appl., 21, 629-635, 1999.

APPENDIX A. BACKGROUND THEORY

A.1 GENERALIZED INVERSE TERMINOLOGY

We summarize below terminology concerning the generalized inverses that are used in the development

of Sections 3 through 5. For a complete coverage of the subject the monograph by Rao and Mitra [21]

may be consulted.

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A.1.1 Pseudo and Spectral G-Inverses of Nondefective Square Matrices

Consider a n n square real matrix A, which may be unsymmetric and singular but is assumed non-defective (that is, it has a complete eigensystem). Its pseudo-inverse, also called the Moore-Penrose

generalized inverse or g-inverse, is the matrix X that satisfies the four Penrose conditions

AXA = A, XAX = X, AX = (AX)T

, XA = (XA)T

. (70)

The pseudo-inverse is identified as X = A+ below. It can be shown that this matrix exists and is unique[10]. The equivalent definition is

AX = PA, XA = PX (71)where PA and PX are projection operators associated with the column space ofA and X, respectively.

IfA is symmetric, so is X and PA = PX = P.The spectral generalized inverse ofA, or simply sg-inverse, is the matrix A that has the same eigenvec-

tors as A, and whose nonzero eigenvalues are the reciprocals of the corresponding nonzero eigenvalues

ofA. (The name and notation for this class of g-inverses is not standardized in the literature; see Rao

and Mitra [21, Sec 4.7] for other identifiers.) More precisely, if the nonzero eigenvalues ofA are i and

associated left and right bi-orthonormalized eigenvectors are xi and yi , respectively, we have

A =

i

i vi yTi , A

=

i

1

ivi y

Ti , i = 0, vTi yj = i j . (72)

where i j is the Kronecker delta. IfA is symmetric, A+ = A. For unsymmetric matrices these two

g-inverses generallydiffer. It canbe shown [21] that A always satisfies the first two Penrose conditions

(70) but not necessarily the others. Note, however, that A, A+ and A have the same rank.

A.1.2 G-Inverses of Rectangular Matrices, Products and Sums

The definitionof pseudo inverse maybe extended to rectangular matrices with real or complex elements.

The general expressions for rectangular real A are

A+ = (ATA)+ AT = AT(AAT)+, (AT)+ = (AAT)+ A = A(ATA)+. (73)This reduces the problem to the pseudo inversion of a symmetric matrix. (For complex matrices

transposes are replaced by conjugate-transposes.) On the other hand, the definition (72) of sg-inverse

is restricted to square matrices.

Let A and B be arbitrary rectangular matrices such that AB is defined. Let B1 = A+A B and A1 =A B1B

+1 . Then AB = A1B1 and (AB)+ = B+1 A+1 [22]. Of particular interest is the case of symmetric

squarerealA and rectangularrealB withthe followingcommon-projectorproperty: A+A = P, PB = B,BB+ = P. Then B1 = PB = B, A1 = AP = A and (AB)+ = B+A+. Applying this to the congruentialtransformation BTAB yields

(BTAB)+ = B+A+(B+)T, (74)In particular, (BTAB)+ = BA+BT ifBTB = I, which can be verified directly.The pseudo inverse of a sum of real matrices A =i Ai is generally a complicated function of the Ai[23]. There is, however, a simple orthogonality result [21, p. 67]:

A+ =

i

A+i if Ai ATj = 0 and ATj Ai = 0, i = j. (75)

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A.2 THE INVERSION OF MODIFIED MATRICES

This section summarizes the major results concerning the inversion of a general square matrix modified

by a lower rank update. For a more thorough coverage of the subject the reader is referred to the survey

articles by Hager [24] or Henderson and Searle [25], and references therein.

A.2.1 The Woodbury Formula

Let A be a square nonsingular n n matrix, the inverse of which is available. This will be called thebaseline matrix. Let U and V be two n kmatrices of rankk, with 1 k n, and S be an invertiblekkmatrix. Assume that a nonsingular kkmatrix T, which satisfies T1+S1 = VT A1 U, exists.The inverse of the baseline matrix A modified by A = U S VT is given by the identity

A1 = (A + A)1 =A U S VT1 = A1 A1 U T VT A1, (76)

This is known as the Woodbury formula [26]. The notation used here is that of Householder [27,

p. 124]. Equation (76) is called the Sherman-Morrison formula [28] ifk= 1, in which case U and V arecolumn vectors and S and T reduce to scalars. The historical development of these important formulas

is discussed in the aforementioned surveys [24,25].

The Woodbury formula can be verified by direct multiplication. A less well known way to prove it

makes use of the block-bordered systemA U

VT S1

B

G

=

In0

(77)

in which In is the n n identity matrix. Elimination in the order G B gives (A U S VT)B = Inwhence B = A1 . Elimination in the order B G and backsubstitution for B gives (76). Note thatG = TVTA1. Both (76) and (77) are of interest to derive efficient computational methods for A1when A is a sparse matrix that can be factored without pivoting, e.g. a symmetric positive-definite

matrix.

A.2.2 Singular Baseline Matrix

Suppose next that A is singular with rankn k, whereas A = A + A = A USVT has fullrankn. If so the Woodbury formula (76) fails, and extensions to cover pseudo-inverses [23,29] are too

complicated to be of practical use.

Although (77) works in principle for singular A because the bordered coefficient matrix is nonsingular,

partial or full pivoting would be necessary in the forward factorization pass. This hinders the computa-

tional efficiency ifA is sparse. It is possible to restore efficiency as follows. The singular A is rendered

nonsingular by adding and subtracting a diagonal matrix DH = HHT, of rankk:

A = A + HHT HHT U S VT = A +HHT [ U H ]

S1 00 Ik

VH

T, (78)

where Ik is the k k identity matrix. The Woodbury formula is applied with A + HHT as baselinematrix. The equivalent block-bordered form is

A+ HHT U HVT S1 0HT 0 Ik

B

GUGH

=

In0

0

(79)

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Because DH = HHT is diagonal, it does not affect the sparseness ofA. IfA is symmetric and A+DHpositive definite, the coefficient matrix in (79) may be stably factored without pivoting. In fact, it is not

even necessary to form A + HHT explicitly because the diagonal entries ofH can be inserted on thefly as explained in Section 7. Upon backsubstitution A1 appears in B.

A.3 THE INVERSION OF MODIFIED SYMMETRIC MATRICES

This final section specializes the results of Section A.2 to symmetric baseline matrices modi fied by a

lower rank symmetric update. The case of singular baseline matrix, which is relevant to the methods of

Section 7, is covered in more detail.

A.3.1 Invertible Baseline Matrices

If the n n baseline matrix A is symmetric and invertible, and the update is A = +UUT, where U isn kand has rankk, we only need to set S = Ik and V = U in (77). The Woodbury formula becomes(A+UUT)1 = A1 A1 U T UT A1, where T1 = UTA1U+ Ik. The explicit inversions can beavoided by solving three linear systems and combining the results:

AX = In, AY = U,YTU + Ik

ZT = YT, A1 = X YZT. (80)

This sequence requires solving for n+kright-hand sides with thenn coefficientmatrix A, andn right-hand sides with the k kcoefficient matrix YTU+ Ik, which is symmetric because YTU = UTA1U.This has computational advantages ifA is sparse and k


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A second useful specialization is U H, the rank regularization matrix introduced in Section A.2.2.From HT(A+HHT)1 H = HTXH = Ikand (84)it isnotdifficult to find the following relations, whichhold for any H as long as rank(J) = k. For notational brevity introduce YH = (A+HHT)1H = XHand YR = (A +HHT)1R = XR. Then

HTYH=

Ik, YT

H

R=

HTYR , YH HT R

=R, YT

H

RRT

=YT

H

,

HTYR RT = YTH, YTHYH = (HTRRTH)1, YH(YTHYH)1YTH = RRT.

(85)

A.3.3 Computing B with Linear Solvers

We are now ready to tackle the efficient computation ofB in

(A +HHTHHT+RRT) B = P, (86)

where P = I RRT. IfA is K or F, B gives F or K, respectively; the first case being more importantin practice. Using A +HHT as baseline matrix, the equivalent block bordered system is

A + HH

T

R HRT Ik 0HT 0 Ik

B

GRGH

=

P0

0

(87)

Applying the staged solving sequence (80) while accounting for the presence ofP on the right gives

(A+ HHT) X = I, (A +HHT) YR = R, (A +HHT) YH = H, (88)YTRR + Ik YTRH

YTHR YTHH Ik

ZTRZTH

=

YTRYTH

, (89)

B =X YRZTR YHZTH

P. (90)

This sequence may be considerably simplified by taking advantage of the relation catalog (85). SinceYTHH = Ik the lower diagonal block of the coefficient matrix of (89) vanishes. The second matrixequation becomes YTHR Z

TR = YTH. Comparing this to the fourth of (85) shows that ZTR = RT, whence

the term YRZTRP in (90) drops out since R

TP = 0. Premultiplying the first matrix equation of (89) byYTH and solving for the remaining unknown gives Z

TH = HRRTHYTH(RYR I) = HTR(YTR RT).

Substitution into (90) and use of the last of (85) yields

B = P X P = P(A + HHT)1P = X YRRT RYTR +RRTYRRT. (91)

This shows that the solution YH of the third linear system in (88) is not actually needed. In fact, the

only place where H appears is in the regularization ofA to AH.

The final result (91) holds for arbitrary H such that J=

HTR has rankk. (It is not restricted to diagonal

HHT, which is used in the actual computations because it preserves sparsity.) In particular, ifH = Rwe recover the third forms listed in (16) and (17). In this special case one of the P multipliers may

be dropped because of (84). But for general H the result is at first sight puzzling. The spectrum of

X = (A + HHT)1 does not look at all like that ofA because the null and range eigenspaces R and VofA become strongly entangled via H, and zero eigenvalues disappear. So how come B = P X P?The answer to the puzzle is (82): the range eigenspace V is imprinted in the spectral morass ofX. Two

projector applications disentagle V and restore the null space.

the construction of free-free flexibility matrices

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